J.P. Morgan Chase & Co. Chief Executive James Dimon and a former top lieutenant recently agreed to a multimillion-dollar resolution of a hiring dispute that entangled the largest U.S. lender, a giant private-equity firm and two longtime fixtures on Wall Street.

Frank Bisignano, a once-trusted ally of Mr. Dimon who left the bank last April to run a payment processor owned by private-equity firm KKR & Co., agreed along with KKR that his new firm will pay millions of dollars to J.P. Morgan, according to people...